## A Classification of C-representations With No (rho,J)-invariant Subspaces

**Point of post: **In this post we shall classify, up to equivalence, the set of all -representations of a finite group . This will enable us, due to a previous characterization, the set of all irreducible -characters of a finite group in terms of the irreducible -characters.

*Motivation*

Clearly our goal in these last few posts was to see how much information about irreducible -representations we could glean from our knowledge of irreducible -representations in the sense that if we know everything we could possibly want to know about the latter for a specific group what could we say about the former. We began this search by showing that there was a bijection between equivalency classes of -representations satisfying the real condition with realizer and no non-trivial proper -invariant subspaces and equivalency classes of irreducible -representations where the class in the domain was mapped to the class in the codomain. We then showed that there was a natural way to create these special -representations from irreducible representations which were either complex or quaternionic. What we shall now do in this post is classify all of these special -representations by showing that, up to equivalence, any such one is either a real irreducible -representation or of the form of one of the representations created by a complex or quaternionic irrep. This will allow us then, since characters are invariant under equivalence, to ascertain all the irreducible -characters of the group in terms of the irreducible -characters.

*Classification of -representations Satisfying the Real Condition With Realizer and having no non-trivial proper -invariant Subspaces*

* *

For the proof that follows allow to be the set of all -representations on satisfying the real condition where, if the realizer is , there does not exist any non-trivial proper -invariant subspaces. With this in mind:

**Theorem: ***Let be a finite group and with realizer . Then, either is a real irreducible -representation or there exists a complex or quaternionic irrep and a complex conjugate such that ( in accordance with the construction in the last theorem).*

**Proof: **Since being a representation of which satisfies the real condition with no non-trivial proper -invariant subspaces we may then assume that is the representation where is an irrep for and the complex conjugate on which is realizer for . Now, if then and is an irrep which must be real since it must commute with (of course we are appealing to our previous characterization of self-conjugate irreps). Now, suppose that . We claim that if then is a -invariant subspace. Indeed, this is clear, namely if then for every

and this last term is clearly in . Since and were arbitrary it follows that is -invariatn as claimed, and since is evidently -invariant, -invariance follows. But, by assumption this implies that from where it follows that if and otherwise. So, we are restricting our attention to representations of the form . Let, and and the projections on these subspaces respectively. Note from the above we’ve determined that . Consider then the mapping . This clearly must be a non-zero transformation since there exists some such that , and so where and so . Moreover, there is an interesting relationship between the map for each . Namely, for any

From this, it follows that if is a complex conjugate on then we have

Thus, if is given by then is an intertwinor for and and since is non-zero by prior discussion we may conclude by Schur’s lemma that . Thus, . The conclusion follows.

**References:**

1. Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Mathematical Society, 1996. Print.

[…] We know from our result concerning the bijection of with that there exists some so that . But, we know that for each one has that is equivalent to where is some real irreducible -representation, […]

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